Table of Content

• What is Euclidean Distance?
• Euclidean Distance Formula
• Euclidean Distance in 3D
• Euclidean Distance in nD
• Euclidean Distance Formula Derivation
• Euclidean Distance and Manhattan Distance
• Solved Questions on Euclidean Distance
• Practice Problems on Euclidean Distance

Aspect	Euclidean Distance	Manhattan Distance
Definition	Measures the shortest straight-line distance between two points.	Measures the distance between two points along axes at right angles.
Formula (2D)	d = √[(x2 – x1)2 + (y2 – y1)2]	d = [|x2 – x1| + |y2 – y1|]
Path	Direct straight line.	Path that resembles city blocks or a grid pattern.
Metric Name	L2 norm or Euclidean norm.	L1 norm or Manhattan norm.
Use Cases	Used in scenarios where direct distances are needed (e.g., physics).	Commonly used in planning, urban design, and certain optimization algorithms.
Sensitivity to Scaling	Less sensitive to scaling of axes.	More sensitive to scaling of axes since it adds absolute differences.

Conclusion

Euclidean Distance is a metric for measuring the distance between two points in Euclidean space, reflecting the length of the shortest path connecting them, which is a straight line. The formula for calculating Euclidean Distance depends on the dimensionality of the space. In a 2-dimensional plane, the distance d between points is, d = d = √[(x₂ – x₁)² + (y₂ – y₁)²]. In 3D, d = √[(x₂ – x₁)² + (y₂ – y₁)²+ (z₂ – z₁)²].

Read More,

• Distance Formula
• 3D Distance Formula
• Section Formula
• Cartesian Coordinate System

Solved Questions on Euclidean Distance

Here are some sample problems based on the distance formula.

Question 1: Calculate the distance between the points (4,1) and (3,0).

Solution:

Using Euclidean Distance Formula:

⇒ d = √(x₂ – x₁)² + (y₂ – y₁)²

⇒ d = √(3 – 4)² + (0 – 1)²

⇒ d = √(1 + 1)

⇒ d = √2 = 1.414 unit

Question 2: Show that the points A (0, 0), B (4, 0), and C (2, 2√3) are the vertices of an Equilateral Triangle.

Solution:

To prove that these three points form an equilateral triangle, we need to show that the distances between all pairs of points, i.e., AB, BC, and CA, are equal.

Distance between points A and B:

AB = √[(4– 0)² + (0-0)²]

⇒ AB = √16

AB = 4 unit

Distance between points B and C:

BC = √[(2-4)^2 + (2√3-0)^2]

⇒ BC = √[4+12] = √16

BC = 4 unit

Distance between points C and A:

CA = √[(0-2)² + (0-2√3)²]

⇒ CA = √[4 + 12] = √16

CA = 4 unit

Here, we can observe that all three distances, AB, BC, and CA, are equal.

Therefore, the given triangle is an Equilateral Triangle

Question 3: Mathematically prove Euclidean distance is a non negative value.

Solution:

Consider two points (x₁, y₁) and (x₂, y₂) in a 2-dimensional space; the Euclidean Distance between them is given by using the formula:

d = √(x₂ – x₁)² + (y₂ – y₁)²

We know that squares of real numbers are always non-negative.

⇒(x₂ – x₁)² >= 0 and (y₂ – y₁)² >= 0

⇒ √(x₂ – x₁)² + (y₂ – y₁)² >= 0

As square root of a non-negative number gives a non-negative number,

Therefore Euclidean distance is a non-negative value. It cannot be a negative number.

Question 4: A triangle has vertices at points A(2, 3), B(5, 7), and C(8, 1). Find the length of the longest side of the triangle.

Solution:

Given, the points A(2, 3), B(5, 7), and C(8, 1) are the vertices of a triangle.

Distance between points A and B:

AB = √[(5-2)² + (7-3)²]

⇒ AB = √9+16= √25

AB = 5 unit

Distance between points B and C:

BC = √[(8-5)² + (1-7)²]

⇒ BC = √[9+36] = √45

BC = 6.708 unit

Distance between points C and A:

CA = √[(8-2)² + (1-3)²]

⇒ CA = √[36+4] = √40

CA = 6.325 unit

Therefore, the length of the longest side of triangle is 6.708 unit.

Practice Problems on Euclidean Distance

P1: Calculate the Euclidean distance between points P(1, 8, 3) and Q(6, 6, 8).

P2: A car travels from point A(0, 0) to point B(5, 12). Calculate the distance traveled by the car?

P3: An airplane flies from point P(0, 0, 0) to point Q(100, 200, 300). Calculate the distance traveled by the airplane.

P4: A triangle has vertices at points M(1, 2), N(4, 6), and O(7, 3). Find the perimeter of the triangle.

P5: On a graph with points K(2, 3) and L(5, 7), plot these points and calculate the Euclidean distance between them.

P6: A drone needs to fly from point A(1, 1) to point B(10, 10). Find the shortest path the drone should take to conserve battery?

P7: A robotic arm moves from position J(1, 2, 3) to position K(4, 5, 6). Calculate the total distance traveled by the robotic arm.

Euclidean Distance – FAQs

Define Euclidean Distance.

Euclidean distance measures the straight-line distance between two points in Euclidean space.

What is the distance formula for a 2D Euclidean Space?

Euclidean Distance between two points (x₁, y1) and (x₂, y₂) in using the formula:

d = √[(x₂ – x₁)² + (y₂ – y₁)²]

What are some properties of Euclidean Distance?

• Euclidean distance is always non-negative because it represents a physical distance in space, which cannot be a negative value.
• Distance between a point and itself is always zero

Can Euclidean Distance be negative?

Euclidean Distance can’t be negative as it represents a physical distance. It can be a zero value but can’t be a negative value.

How can Euclidean Distance be extended to higher dimensions?

In general, the Euclidean Distance formula between two points (x₁₁, x₁₂, x₁₃, …., x_1n) and (x₂₁, x₂₂, x₂₃, …., x_2n) in an n-dimensional space is given by the formula:

d = √[∑(x_2i – x_1i)²]

What is the difference between Euclidean Distance and Manhattan Distance?

Consider two points (x_1, y₁) and (x₂, y₂) in a 2-dimensional space;

Euclidean Distance between them is given by using the formula:

d = √[(x₂ – x₁)² + (y₂ – y₁)²], (Calculates the square root of the sum of squared differences)

Manhattan Distance between them is given by using the formula:

d = [|x₂ – x_1| + |y₂ – y₁|], (Calculates the distance between two points as the sum of the absolute differences in their coordinates)